Magnetohydrodynamic state estimation with boundary sensors

We present a PDE observer that estimates the velocity, pressure, electric potential and current fields in a magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow. This flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier–Stokes equations and a Poisson equation for the electric potential under the so-called inductionless MHD approximation in a low magnetic Reynolds number regime. We identify physical quantities (measurable on the wall of the channel) that are sufficient to generate convergent estimates of the velocity, pressure, and electric potential field away from the walls. Our observer consists of a copy of the linearized MHD equations, combined with linear injection of output estimation error, with observer gains designed using backstepping. Pressure, skin friction and current measurements from one of the walls are used for output injection. For zero magnetic field or nonconducting fluid, the design reduces to an observer for the Navier–Stokes Poiseuille flow, a benchmark for flow control and turbulence estimation. We show that for the linearized MHD model the estimation error converges to zero in the L 2 norm. Despite being a subject of practical interest, the problem of observer design for nondiscretized 3-D MHD or Navier–Stokes channel flow has so far been an open problem.